Fields with large Kronecker constants

For any algebraic number field K there is a positive number ? such that if α is a nonzero integer of K other than a root of unity, then at least one conjugate of α has absolute value ≥ 1 + ?. It has been conjectured that ? can be taken as $\text{2}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{? 1}$, where n is the degree of K over the field of rationals. In this paper various conditions are discussed under which the validity of this conjecture can be established.